Simplicial Volume and Scalar Curvature on Closed K\"ahler Surfaces

Abstract

Let $M$ be a closed K\"ahler surface. We prove that every Riemannian metric $g$ on $M$ with $\operatorname{Sc}_g\geq-\lambda^2$, where $\lambda\geq 0$, satisfies $$ \lVert M\rVert\leq \frac{27}{2}\,\lambda^4\operatorname{vol}_g(M). $$ This proves Gromov's quantitative scalar-curvature--simplicial-volume conjecture for closed K\"ahler surfaces. We also construct infinitely many non-K\"ahler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.